Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

If \[\vec{a}\] and \[\vec{b}\] are unit vectors, then find the angle between \[\vec{a}\] and \[\vec{b}\] given that \[\left( \sqrt{3} \vec{a} - \vec{b} \right)\] is a unit vector.      

Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].

 If the line \[\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}\]  lies in the plane  \[lx + my - z =\]   then find the value of  \[l^2 + m^2\] .

Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .

Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] are coplanar if and only if \[\vec{a} + \vec{b}\], \[\vec{b} + \vec{c}\] and \[\vec{c} + \vec{a}\] are coplanar.

If  \[\vec{a} \text{ and } \vec{b}\] are two non-collinear unit vectors such that \[\left| \vec{a} + \vec{b} \right| = \sqrt{3},\] find \[\left( 2 \vec{a} - 5 \vec{b} \right) \cdot \left( 3 \vec{a} + \vec{b} \right) .\] 

Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx`  and hence evaluate   `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .   

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.

Find : `int_  (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.

Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`

Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`

Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`

Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`

Evaluate `int_(-1)^2 "f"(x)  "d"x`, where f(x) = |x + 1| + |x| + |x – 1|

`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.

`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.

If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.

`int_(-2)^2 |x cos pix| "d"x` is equal to ______.

`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.